3 ABSOLUTELY CONVERGENT SERIES 97

Show that if lim r(n) = 0, the series (x^ is convergent in E and that E •*» = E x*w •

n n

(Evaluate the difference E **<n)) — E x* f°r large n.)

fc = 0 fc=0

4. Let (xmn) (m ^ 0, n ^ 0) be a double sequence of points of a normed space E. Suppose
that: (1) for each m^Q, the series xmQ + jcmi + • • • + xmn + • - • is convergent in E;

let .xm be its sum, and let rmn = xmn +xmttt+1 -\-----; (2) for each n *z 0, the series

r0n + fin H-----+ rmn H-----is convergent in E; let tn be its sum.

(a) Show that for each n ^ 0, the series x0n-\- xln-\-----+ xmn -\-----is convergent; let

zn be its sum.

oo oo

(b) In order that E Vm ~ ^zn,itis necessary and sufficient that lim tn = 0.

5. (a) Show that the series E —2------2 *s convergent and has a sum equal to — 3/4m2

(decompose the rational fraction l/(m2 — x2)).

(b) Let umn — —------- if m ^ n, and unn = 0; show that

m2 — H2

00/00 \ 00/00 \

E Ztfm.)« - El Z«m« ^0.

m=0\n=0 / n=0\m=0 /

6. If / is a function defined in N x N, with values in a metric space, we denote by
lim f(m, n) the limit of / (when it exists) at the point (+ oo, 4- °°) of R x R, with

m-+oo,n-*oo

respect to the subspace N x N (Section 3.13). Let (jcwn) be a double sequence of real
numbers, and let smn = E *** •

h^m, k^n

(a) If lim smn exists, then lim xmn = 0. Give an example in which xnm = xmn,

jn-n»,n-»oo m-npo,n-»QD

Xm, 2n= -*m,2n+l = ~^m4-l,2« for W ^ 2/2 + 1, X2n, 2n = 0, SUCh that Hm ^mn == 0,

m-*oo,n-*-oo

and none of the series xm0 -f ^mi H---------h xmw H------, *0n + -^in H------+ xmn -\------is

convergent.

(b) Give an example in which xmn = 0 except if m = n+ l,m = n or n = mjrl

00 00 00 00

(hence all series E •*««» E ^wn are convergent), E *»« = E x'"« ~ ^ ^or a^ indices

«i=!0 m = 0 n = 0 m = 0

m, «, but lim smn does not exist.

3. ABSOLUTELY CONVERGENT SERIES

(5.3.1) In order that a series (xn) of positive numbers be convergent it is
necessary and sufficient that for a strictly increasing sequence (kn) of integers

oo

^0, the sequence (skn) of partial sums be majorized, and then the sum s = 5] xn

n = 0

is equal to sup skn.

n

The assumption xn ^ 0 is equivalent to 5(n_1 ^^n, and then the result
follows at once from (4.2.1).dimen-